Log Likelihood, Part 1

To compute the log likelihood, we need to get the ratios and use them to compute a score that will allow us to decide whether a tweet is positive or negative. The higher the ratio, the more positive the word is:

To do inference, you can compute the following:

$\frac{P(p o s)}{P(n e g)} \prod_{i=1}^{m} \frac{P\left(w_{i} \mid p o s\right)}{P\left(w_{i} \mid n e g\right)} >1$

As $m$ gets larger, we can get numerical flow issues, so we introduce the $\log$, which gives you the following equation:

$\log \left(\frac{P(p o s)}{P(n e g)} \prod_{i=1}^{n} \frac{P\left(w_{i} \mid p o s\right)}{P\left(w_{i} \mid n e g\right)}\right) \Rightarrow \log \frac{P(p o s)}{P(n e g)}+\sum_{i=1}^{n} \log \frac{P\left(w_{i} \mid p o s\right)}{P\left(w_{i} \mid n e g\right)}$

The first component is called the log prior and the second component is the log likelihood. We further introduce $\lambda$ as follows:

Having the $\lambda$ dictionary will help a lot when doing inference.